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Unformatted text preview: 1 COMP170 Discrete Mathematical Tools for Computer Science Discrete Math for Computer Science K. Bogart, C. Stein and R.L. Drysdale Section 3.2, pp. 104-114 Quantifiers Version 2.0: Last updated, May 13, 2007 Slides c 2005 by M. J. Golin and G. Trippen 2 3.2 Variables and Quantifiers • Variables and Universes • Quantifiers • Standard Notation for Quantification • Statements about Variables • Proving Quantified Statements True or False • Negation of Quantified Statements • Implicit Quantification 3 Consider the statement: Variables and Universes (*) m 2 &gt; m Is (*) True or False ? This is an ill-posed question! For some values of m , e.g., m = 2 , (*) is True For other values of m , e.g., m = 1 / 2 , (*) is False In statements such as m 2 &gt; m , variable m is not constrained . Unconstrained variables are called free variables . Each possible value of a free variable gives a new statement. The Truth or Falsehood of this new statement, is determined by substituting in the new value for the variable. 4 • For which values of m is (*) True and for which values is it False ? • For the universe of non-negative integers , the statement is True for every value of m except m = 0 , 1 . • For the universe of real numbers , the statement is True for every value of m except for ≤ m ≤ 1 Two main points: • Clearly state the universe • A statement about a variable can be True for some values of a variable and False for others. Again consider the statement: (*) m 2 &gt; m • This statement is also ill-defined! The answer depends upon which universe we assume 5 3.2 Variables and Quantifiers • Variables and Universes • Quantifiers • Standard Notation for Quantification • Statements about Variables • Proving Quantified Statements True or False • Negation of Quantified Statements • Implicit Quantification 6 Quantifiers The statement • While m 2 &gt; m is True for values such as m =- 3 or m = 9 it is False for m = 0 or m = 1 . • Thus, it is not True that m 2 &gt; m for every integer m , so (*) is False (*) For every integer m , m 2 &gt; m is False . 7 Quantifiers The statement • A phrase like for every integer m that converts a symbolic statement about potentially any member of our universe into a statement about the universe is called a quantifier • A quantifier that asserts a statement about a variable is true for...
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